description/proof of that for \(C^\infty\) manifold with boundary and real vectors space at each point with same dimension, \(k\), to-be-set-of-local-trivializations determines \(C^\infty\) vectors bundle of rank \(k\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle of rank \(k\).
- The reader knows a definition of %category name% isomorphism.
- The reader knows a definition of general linear group of vectors space.
- The reader admits the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas.
- The reader admits the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.
- The reader admits the proposition that any map between arbitrary subsets of any \(C^\infty\) manifolds with boundary bijective and locally diffeomorphic at each point is a diffeomorphism.
- The reader admits the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary and any real vectors space at each point with any same dimension, \(k\), any to-be-set-of-local-trivializations determines the canonical \(C^\infty\) vectors bundle of rank \(k\).
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(k\): \(\in \mathbb{N} \setminus \{0\}\)
\(\{V_m \in \{\text{ the } k \text{ -dimensional } \mathbb{R} \text{ vectors spaces }\} \vert m \in M\}\):
\(E\): \(= \biguplus \{V_m\}\), where \(\biguplus\) denotes the disjoint union
\(\pi\): \(: E \to M, v \in V_m \mapsto m\)
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{U_j \in \{\text{ the open subsets of } M\} \vert j \in J\}\): such that \(\cup \{U_j\} = M\)
\(\{\Phi_j: \pi^{-1} (U_j) \to U_j \times \mathbb{R}^k \vert j \in J\}\), such that for each \(m \in U_j\), \(\Phi_j \vert_{V_m}: V_m \to \{m\} \times \mathbb{R}^k \in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\}\)
//
Statements:
\(\exists \{(U_m \subseteq M, \phi_m) \in \{\text{ the charts for } M\} \vert \exists j \in J (m \in U_m \subseteq U_j)\} \in \{\text{ the countable sets }\}\), such that \(\cup \{U_m\} = M\)
\(\land\)
(
(
\(\forall j, l \in J \text{ such that } U_j \cap U_l \neq \emptyset (\exists T_{j, l}: U_j \cap U_l \to GL (\mathbb{R}^k) \in \{\text{ the } C^\infty maps \})\)
\(\land\)
\(\Phi_l \circ {\Phi_j}^{-1} \vert_{(U_j \cap U_l) \times \mathbb{R}^k}: (U_j \cap U_l) \times \mathbb{R}^k \to (U_j \cap U_l) \times \mathbb{R}^k, (m, v) \mapsto (m, T_{j, l} (m) v)\)
)
\(\implies\)
(
\(\{(\pi^{-1} (U_m) \subseteq E, \widetilde{\phi_m}) \vert U_m \in \{U_m\}, \widetilde{\phi_m}: \pi^{-1} (U_m) \to \mathbb{R}^k \times \phi_m (U_m) = \lambda \circ (\phi_m, id) \circ \Phi_j \vert_{\pi^{-1} (U_m)}\}\), where \(\lambda: \mathbb{R}^{d + k} \to \mathbb{R}^{d + k}, (x^1, ..., x^d, x^{d + 1}, ..., x^{d + k}) \mapsto (x^{d + 1}, ..., x^{d + k}, x^1, ..., x^d)\), is a to-be-atlas for the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas which makes \(E\) the canonical \(C^\infty\) manifold with boundary, which does not depend on the choice of \(\{U_m\}\)
\(\land\)
\((E, M, \pi) \in \{\text{ the } C^\infty \text{ vectors bundles }\}\)
)
)
//
2: Note
\(\lambda\) is used in order for \(\widetilde{\phi_m} (\pi^{-1} (U_m))\) to be a subset of \(\mathbb{H}^{d + k}\) when \(M\) is with a nonempty boundary: although usually \(\lambda\) is sloppily omitted, \((\phi_m, id) \circ \Phi_j\) without \(\lambda\) would not be any map into \(\mathbb{H}^{d + k}\), because it would make \(0 \le x^d\) and \(- \infty \lt x^{d + k} \lt \infty\).
3: Proof
Whole Strategy: Step 1: see that \(\{(\pi^{-1} (U_m) \subseteq E, \widetilde{\phi_m})\}\) is a to-be-atlas for the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas; Step 2: see that \((E, M, \pi)\) is a \(C^\infty\) vectors bundle; Step 3: see that the topology and the atlas of \(E\) does not depend on the choice of \(\{(U_m \subseteq M, \phi_m)\}\).
Step 1:
We can take \(\{(U_m \subseteq M, \phi_m)\}\) as a countable set, because \(M\) is 2nd-countable: each \(U_m\) can be chosen from a countable basis: while there is a charts cover of \(M\), for each \(m \in M\), \(m\) is contained in the intersection of a chart domain and a \(U_j\), and an element of the basis can be chosen to contain \(m\) and be contained in the intersection of the chart domain and \(U_j\), and \((U_m \subseteq M, \phi_m)\) can be taken to be the restriction of the chart (use a fixed chart for each \(U_m\)). Note that for some \(m, m' \in M\) such that \(m \neq m'\), \(U_m = U_{m'}\) can happen, which is the reason why \(\{U_m\}\) is countable while \(M\) may be uncountable. Note that \(J\) may be uncountable but not all the \(U_j\) s need to be used.
Let us see that \(\{(\pi^{-1} (U_m) \subseteq E, \widetilde{\phi_m})\}\) is a to-be-atlas for the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas.
\(\Phi_j: \pi^{-1} (U_j) \to U_j \times \mathbb{R}^k\) is a bijection: for each \(e, e' \in \pi^{-1} (U_j)\) such that \(e \in V_p\) and \(e' \in V_{p'}\) where \(p \neq p'\), \(\Phi_j (e) \neq \Phi_j (e')\), because \(\Phi_j (e) \in \{p\} \times \mathbb{R}^k\) and \(\Phi_j (e') \in \{p'\} \times \mathbb{R}^k\); for each \(e, e' \in \pi^{-1} (U_j)\) such that \(e, e' \in V_p\) but \(e \neq e'\), \(\Phi_j (e) \neq \Phi_j (e')\), because \(\Phi_j \vert_{V_p}\) is bijective; each point on \(U_j \times \mathbb{R}^k\) is mapped to by \(\Phi_j\), because each point on \(\{p\} \times \mathbb{R}^k\) is mapped to by \(\Phi_j \vert_{V_p}\), because \(\Phi_j \vert_{V_p}\) is bijective.
\(\widetilde{\phi_m} =\lambda \circ (\phi_m, id) \circ \Phi_j \vert_{\pi^{-1} (U_m)}\) is an injection, because \(\Phi_j\), \((id, \phi_m)\), and \(\lambda\) are injections.
\(\widetilde{\phi_m} (\pi^{-1} (U_m)) = \lambda \circ (\phi_m, id) \circ \Phi_j (\pi^{-1} (U_m)) = \mathbb{R}^k \times \phi_m (U_m)\), which is open on \(\mathbb{R}^{d + k}\) or \(\mathbb{H}^{d + k}\).
\(\widetilde{\phi_m} (\pi^{-1} (U_m) \cap \pi^{-1} (U_{m'})) = \mathbb{R}^k \times \phi_m (U_m \cap U_{m'})\) is an open subset of \(\widetilde{\phi_m} (\pi^{-1} (U_m)) = \mathbb{R}^k \times \phi_m (U_m)\).
\(\Phi_l \circ {\Phi_j}^{-1} \vert_{(U_j \cap U_l) \times \mathbb{R}^k}: (U_j \cap U_l) \times \mathbb{R}^k \to (U_j \cap U_l) \times \mathbb{R}^k, (m, v) \mapsto (m, T_{j, l} (m) v)\) is a diffeomorphism: it is bijective, because there is the inverse, \((m, v) \mapsto (m, T_{j, l}^{-1} (m) v)\): \((m, v) \mapsto (m, T_{j, l} (m) v) \mapsto (m, T_{j, l}^{-1} (m) T_{j, l} (m) v) = (m, v)\) and \((m, v) \mapsto (m, T_{j, l}^{-1} (m) v) \mapsto (m, T_{j, l} (m) T_{j, l}^{-1} (m) v) = (m, v)\); it is \(C^\infty\), because \(T_{j, l} (m) v\) is \(C^\infty\) with respect to \(m\) and \(v\); the inverse, \((m, v) \mapsto (m, T_{j, l}^{-1} (m) v)\), is \(C^\infty\), because each component of \(T_{j, l}^{-1}\) is a polynomial of the components of \(T_{j, l}\) divided by the determinant of \(T_{j, l}\), which (the determinant) is a nonzero polynomial of the components of \(T_{j, l}\).
\(\widetilde{\phi_{m'}} \circ {\widetilde{\phi_m}}^{-1} \vert_{\widetilde{\phi_m} (\pi^{-1} (U_m) \cap \pi^{-1} (U_{m'}))}: \widetilde{\phi_m} (\pi^{-1} (U_m) \cap \pi^{-1} (U_{m'})) \to \widetilde{\phi_{m'}} (\pi^{-1} (U_m) \cap \pi^{-1} (U_{m'}))\) is a diffeomorphism: \(= \lambda \circ (\phi_{m'}, id) \circ \Phi_{j'} \circ {\Phi_j}^{-1} \circ ({\phi_{m'}}^{-1}, id^{-1}) \circ {\lambda}^{-1}\), but \(\lambda\), \((id, \phi_{m'})\), \(\Phi_{j'} \circ {\Phi_j}^{-1}\), \((id^{-1}, {\phi_{m'}}^{-1})\), and \({\lambda}^{-1}\) are some diffeomorphisms.
So, we know that \(D\) in the proposition that for any set, any to-be-atlas determines the canonical topology and the atlas is a topological basis and \(E\) is the topological space with the basis.
Let us check that \(E\) is Hausdorff.
Let \(e, e' \in E\) be any such that \(e \neq e'\).
There is a \(U_m \in \{U_m\}\) such that \(\pi (e), \pi (e') \in U_m\) or there is not any such.
Let us suppose that there is such a \(U_m\).
\(e, e' \in \pi^{-1} (U_m)\).
As \(\widetilde{\phi_m} (\pi^{-1} (U_m)) = \mathbb{R}^k \times \phi_m (U_m)\) is Hausdorff, \(\pi^{-1} (U_m)\) is Hausdorff, and so, there are an open neighborhood of \(e\), \(U_e \subseteq \pi^{-1} (U_m)\), and an open neighborhood of \(e'\), \(U_{e'} \subseteq \pi^{-1} (U_m)\), such that \(U_e \cap U_{e'} = \emptyset\). As \(\pi^{-1} (U_m)\) is an open subspace of \(E\), \(U_e\) and \(U_{e'}\) are some open subsets of \(E\), by the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.
Let us suppose that there is no such a \(U_m\).
There are a \(U_m \in \{U_m\}\) such that \(\pi (e) \in U_m\) and a \(U_{m'} \in \{U_m\}\) such that \(\pi (e') \in U_{m'}\). It may not be that \(U_m \cap U_{m'} = \emptyset\), but there are an open neighborhood of \(\pi (e)\), \(U_{\pi (e)} \subseteq M\), and an open neighborhood of \(\pi (e')\), \(U_{\pi (e)'} \subseteq M\), such that \(U_{\pi (e)} \subseteq U_m\), \(U_{\pi (e')} \subseteq U_{m'}\), and \(U_{\pi (e)} \cap U_{\pi (e')} = \emptyset\): as \(M\) is Hausdorff, there are an open neighborhood of \(\pi (e)\), \(V_{\pi (e)} \subseteq M\), and an open neighborhood of \(\pi (e')\), \(V_{\pi (e)'} \subseteq M\), such that \(V_{\pi (e)} \cap V_{\pi (e')} = \emptyset\), and we can take \(U_{\pi (e)} := V_{\pi (e)} \cap U_m\) and \(U_{\pi (e')} := V_{\pi (e')} \cap U_{m'}\).
\(e \in \pi^{-1} (U_{\pi (e)}) \subseteq \pi^{-1} (U_m)\) and \(e' \in \pi^{-1} (U_{\pi (e')}) \subseteq \pi^{-1} (U_{m'})\). \(\pi^{-1} (U_{\pi (e)})\) is an open subset of \(\pi^{-1} (U_m)\), because \(\widetilde{\phi_m} (\pi^{-1} (U_{\pi (e)})) = \mathbb{R}^k \times \phi_m (U_{\pi (e)})\) is an open subset of \(\widetilde{\phi_m} (\pi^{-1} (U_m)) = \mathbb{R}^k \times \phi_m (U_m)\). As \(\pi^{-1} (U_{m'})\) is open on \(E\), \(\pi^{-1} (U_{\pi (e)})\) is open on \(E\), by the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space. Likewise, \(\pi^{-1} (U_{\pi (e')})\) is open on \(E\). \(\pi^{-1} (U_{\pi (e)}) \cap \pi^{-1} (U_{\pi (e')}) = \emptyset\), because \(U_{\pi (e)} \cap U_{\pi (e')} = \emptyset\).
So, \(E\) is Hausdorff.
So, \(E\) is the \(C^\infty\) manifold with boundary with the atlas.
Step 2:
\(\pi\) is \(C^\infty\), because for each \(e \in E\), there are a chart, \((\pi^{-1} (U_m) \subseteq E, \widetilde{\phi_m})\), such that \(e \in \pi^{-1} (U_m)\) and the chart, \((U_m \subseteq M, \phi_m)\), such that \(\pi (\pi^{-1} (U_m)) \subseteq U_m\), and the components function is \(: (x, \phi_m (\pi (e))) \mapsto \phi_m (\pi (e))\), which is \(C^\infty\).
\(\Phi_j \vert_{V_p}\) is a 'vectors spaces - linear morphisms' isomorphism.
\(\Phi_j\) is a diffeomorphism, because it is locally diffeomorphic and is bijective, by the proposition that any map between arbitrary subsets of any \(C^\infty\) manifolds with boundary bijective and locally diffeomorphic at each point is a diffeomorphism: for each \(p \in \pi^{-1} (U_j)\), \(p \in \pi^{-1} (U_m)\), but \(\widetilde{\phi_m}\) may have been constructed from a \(\Phi_{j'}\) not \(\Phi_j\), but \(\Phi_j \vert_{\pi^{-1} (U_j) \cap \pi^{-1} (U_m)}: \pi^{-1} (U_j) \cap \pi^{-1} (U_m) \to (U_j \cap U_m) \times \mathbb{R}^k = \Phi_j \circ {\Phi_{j'}}^{-1} \vert_{(U_j \cap U_m) \times \mathbb{R}^k} \circ \Phi_{j'} \vert_{\pi^{-1} (U_j) \cap \pi^{-1} (U_m)}\), and \(\Phi_{j'} \vert_{\pi^{-1} (U_j) \cap \pi^{-1} (U_m)}: \pi^{-1} (U_j) \cap \pi^{-1} (U_m) \to (U_j \cap U_m) \times \mathbb{R}^k\) is a diffeomorphism, because \(\Phi_{j'} \vert_{\pi^{-1} (U_j) \cap \pi^{-1} (U_m)} = ({\phi_m}^{-1}, id^{-1}) \circ \lambda^{-1} \circ \widetilde{\phi_m} \vert_{\pi^{-1} (U_j) \cap \pi^{-1} (U_m)}\), and \(\Phi_j \circ {\Phi_{j'}}^{-1} \vert_{(U_j \cap U_m) \times \mathbb{R}^k}: (U_j \cap U_m) \times \mathbb{R}^k \to (U_j \cap U_m) \times \mathbb{R}^k\) is a diffeomorphism, so, \(\Phi_j \vert_{\pi^{-1} (U_j) \cap \pi^{-1} (U_m)}\) is a diffeomorphism.
So, \(\Phi_j\) is a local trivialization.
So, \((E, M, \pi)\) is a \(C^\infty\) vectors bundle.
Step 3:
Let us see that the topology and the atlas of \(E\) does not depend on the choice of \(\{(U_m \subseteq M, \phi_m)\}\).
Let us take any other \(\{(V_n \subseteq M, \psi_n)\}\).
\(\{U_m \cap V_n\}\) covers \(M\), because \(M = M \cap M = (\cup_m U_m) \cap (\cup_n V_n) = \cup_{m, n} (U_m \cap V_n)\). \(\{\pi^{-1} (U_m \cap V_n)\}\) covers \(E\).
Each \(U_m \cap V_n\) is an open subset of \(M\) and \(\pi^{-1} (U_m \cap V_n)\) is an open subset of \(E\) in both the topologies: \(\pi\) is continuous anyway. \((\pi^{-1} (U_m \cap V_n) \subseteq E, \widetilde{\phi_m} \vert_{\pi^{-1} (U_m \cap V_n)})\) and \((\pi^{-1} (U_m \cap V_n) \subseteq E, \widetilde{\psi_n} \vert_{\pi^{-1} (U_m \cap V_n)})\) are some charts.
\(\widetilde{\psi_n} \vert_{\pi^{-1} (U_m \cap V_n)} \circ {\widetilde{\phi_m} \vert_{\pi^{-1} (U_m \cap V_n)}}^{-1}: \mathbb{R}^k \times \phi_m (U_m \cap V_n) \to \mathbb{R}^k \times \psi_n (U_m \cap V_n) = \lambda \circ (\psi_n, id) \circ \Phi_l \circ {\Phi_j}^{-1} \circ ({\phi_m}^{-1}, id^{-1}) \circ \lambda^{-1} \vert_{\mathbb{R}^k \times \phi_m (U_m \cap V_n)}\), which is bijective and \(C^\infty\), because \(\lambda\), \((\psi_n, id)\), \(\Phi_l \circ {\Phi_j}^{-1}\), \(({\phi_m}^{-1}, id^{-1})\), and \(\lambda\) are \(C^\infty\). The inverse is \(\widetilde{\phi_m} \vert_{\pi^{-1} (U_m \cap V_n)} \circ {\widetilde{\psi_n} \vert_{\pi^{-1} (U_m \cap V_n)}}^{-1}\), which is \(C^\infty\) likewise. So, \(\widetilde{\psi_n} \vert_{\pi^{-1} (U_m \cap V_n)} \circ {\widetilde{\phi_m} \vert_{\pi^{-1} (U_m \cap V_n)}}^{-1}\) is a diffeomorphism.
By the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same, the 2 topology-atlas pairs are the same.
That means that the topology and the atlas of \(E\) does not depend on the choice of \(\{(U_m \subseteq M, \phi_m)\}\).
